Let's start with drill bit size - that is the size of the drill that the PCB fabricator uses to drill holes in the PCB for mounting holes, vias, and thru-hole pads. They come in standard sizes, so you can't specify just any hole size and expect the fabricator to be able to make it - the hole has to match the available drill sizes.
If you are specifying hole sizes on a PCB, you normally specify "finished" hole size, not the drill size. The finished size of a via hole or thru-hole will be smaller than the drill size. The reason is the copper plating that will be done inside the drilled hole to make the connections to the various layers of the board. Usually, but not always, the finished, plated, hole size is about 2-4mils smaller than the drill size because of the copper plating.
The size of the hole you put in a via or pad is dependent on whether or not you need to solder a lead in the hole, the available space, the frequency of the signal passing through the plated hole, the current, etc. As the designer, you tell the fabricator what size you want the hole when it is plated. The fabricator picks the drill size that will make a hole big enough to give the finished size you want after plating. As designer, you also have the responsibility to make sure that there is enough pad or via diameter to allow the fabricator to choose his drill. If you specify too big a finished hole for the pad size, the fabricator's drill will remove all the pad or via copper. Normally the fabricator's specifications tell you what minimum "annulus" he needs remaining on the pad or via after drilling to ensure he can properly plate the inside of the hole.
The term "via" means a connecting plated hole between two layers. Vias can connect top signal paths to bottom signal paths or they can connect signals on any two other layers. A special case of a via is a "blind via". The blind via doesn't go all the way through the board. It connects two symetrically spaced inner layers together, but doesn't go all the way through to the top and bottom. They are made by drilling and plating holes in the early stages of PCB fabrication as the board layers are laminated to the core in opposing symetric pairs.
"Thru-hole pads" are the same as vias but are usually associated with components - for example the pads for a resistor with leads, a capacitor with leads or an IC with leads. They can also be single "free" pads that connect traces layer-to-layer like a via (free means that it isn't part of a component). From the standpoint of terminology, pads aren't restricted to connecting only two layers like a via. A thru-hole pad can connect on any or all of the layers that it passes through. For example, if I wanted to connect a trace on the top layer to traces on inner layer 2, layer 6, and the bottom layer - I would use a free thru-hole pad. If I wanted to connect a trace on the top layer to a trace on just inner layer 2 - I would use a via.
The distinction between a free pad and a via is just terminology. They are both used to connect between layers. Your PCB editor software is probably programmed to allow a via only to connect two layers, but it will allow a thru-hole pad to connect many layers.